A016186 Expansion of 1/((1-8*x)*(1-10*x)).

1, 18, 244, 2952, 33616, 368928, 3951424, 41611392, 432891136, 4463129088, 45705032704, 465640261632, 4725122093056, 47800976744448, 482407813955584, 4859262511644672, 48874100093157376, 490992800745259008, 4927942405962072064, 49423539247696576512, 495388313981572612096, 4963106511852580896768

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is the number of strings of n+1 decimal digits having an odd number of 0's. For 2 digits these are for example the 18 strings 01, 02, 03, ..., 09, 10, 20, 30, ..., 90. - Geoffrey Critzer, Jan 24 2011

Links

• G. C. Greubel, Table of n, a(n) for n = 0..990
• Index entries for linear recurrences with constant coefficients, signature (18,-80).

Crossrefs

Cf. A016140, A060531, A081035, A081203.

Programs

Magma

[2^n*(5^(n+1)-4^(n+1)): n in [0..40]]; // G. C. Greubel, Nov 14 2024

Mathematica

Rest@With[{m=30}, CoefficientList[Series[Exp[9 x] Sinh[x], {x,0,m}], x]*Range[0, m]!]
Table[2^n*(5^(n+1)-4^(n+1)), {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
LinearRecurrence[{18,-80},{1,18},30] (* Harvey P. Dale, Aug 26 2019 *)

PARI

Vec(1/((1-8*x)*(1-10*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

SageMath

A016186=BinaryRecurrenceSequence(18,-80,1,18)
print([A016186(n) for n in range(41)]) # G. C. Greubel, Nov 14 2024

Formula

From R. J. Mathar, Sep 18 2008: (Start)

a(n) = 5*10^n - 4*8^n = A081203(n+1).

Binomial transform of A081035. (End)

From Geoffrey Critzer, Jan 24 2011: (Start)

a(n) = 8*a(n-1) + 10^(n-1).

E.g.f.: exp(9*x)*sinh(x) (with offset 1). (End)

A060531(n) = a(n) - 9*a(n-1). - R. J. Mathar_, Jan 27 2011

From Vincenzo Librandi, Feb 09 2011: (Start)

a(n) = 10*a(n-1) + 8^n, a(0)=1.

a(n) = 18*a(n-1) - 80*a(n-2), a(0)=1, a(1)=18. (End)

E.g.f.: exp(9*x)*( cosh(x) + 9*sinh(x) ). - G. C. Greubel, Nov 14 2024

Extensions

More terms added by G. C. Greubel, Nov 14 2024